B Dlν and B πlν on the Lattice

B Dlν and B πlν on the Lattice Paul Mackenzie Fermilab mackenzie@fnal.gov Thanks, Ruth van de Water, Richard Hill, Thomas Becher BaBar/Lattice QCD Workshop SLAC Sept. 16, 006 1

Lattice calculations Quarks are defined on the sites of the lattice, and gluons are SU3 matrices on the links, U=exp(igA). Calculations are done at finite lattice spacing a, finite volume V, too high a light quark mass m, etc. Theory is used to derive expected functional form for extrapolations in a (OPE), m (chiral PT), V (finite volume chiral PT)... Systematic errors.

Progress for simple quantities Quantities that used to agree decently, ~10%, in the quenched approximation...... agree to a few % in recent unquenched calculations. Gold-plated quantities. Staggered fermions, the least CPU-intensive. Paul Mackenzie Fermilab Wine and Cheese, July 14, 006 3 3

Three families of lattice fermions Staggered (Kogut-Susskind)/naive Good chiral behavior (can get to light quark masses). Fermion doubling introduces nonlocal effects which must be theoretically understood. Cheap. Wilson/clover No fermion doubling but horrible chiral behavior. Overlap/domain wall Nice chiral behavior at the expense of adding a fifth space-time dimension. Expensive. Staggered fermion unquenched calculations are the cheapest and currently most advanced phenomenologically, (probably a temporary situation). Paul Mackenzie Fermilab Wine and Cheese, July 14, 006 4 4

Gold-plated quantities of lattice QCD Quantities that are easiest for theory and experiment to both get right. Stable particle, one-hadron processes. Especially mesons. More complicated methods are required for multihadron processes: - unstable particles are messy to interpret, - multihadron final states are different in Euclidean and Minkowski space. Paul Mackenzie Fermilab Wine and Cheese, July 14, 006 5 5

Many of the most important quantities for lattice QCD are golden quantities. E.g., measurements determining the fundamental parameters of the Standard Model. Paul Mackenzie Fermilab Wine and Cheese, July 14, 006 6 6

B Dlν dγ dω (B D( ) ) V cb F B D ( )(ω) Form factors are well described by the Isgur-Wise function. Governed by two parameters to good approximation: normalization and slope. Slope parameter is well measured by experiment. = To obtain Vcb from data, theory must supply only normalization, which can be obtained from B V 0 D at zero recoil. 7 7

B Dlν Z B V 0 D Ratio method: determine from a ratio that goes to 1 with vanishing errors in the symmetry limit. Hashimoto et al. (99), (Works for K πlν, too, Becirevic et al.) C DV 0 B (t)c BV 0 D (t) C DV 0 D (t)c BV 0 B (t) D V 0 B B V 0 D D V 0 D B V 0 B Uncertainties cancel in ratio in the symmetry limit. Used in renormalization of the vector current. 1.1 F B D (1) = 1.074 (18) sta (15) sys F(1) B!>D N f =+1 (FNAL/MILC) N f =0 (FNAL 99) Using HFAG 04 avg for V cb F (1), V cb Lat05 =3.91(09) lat (34) exp 10 1 0 0.01 0.0 0.03 m l F ( ) = ( ) ( ) Fermilab/MILC 05. Okamoto, Lattice 005 8 8

s f - e- is e F + - y I o B πlν 10 BABAR 5 I also review the form factors in other processes. Some of the recent work on the lattice QCD calculations of the B πlν B ρlν form factors in relativistic formalism are presented. Very precise calculations of semileptonic form factors for B D ( ) lν at zero recoil and the calculations of 0 0 10 0 q Data has nontrivial shape. the slope of the Isgur Wise function are presented. B πlν Theory and experimental uncertainties are q dependent, severely so on the lattice. Harder and more important to understand shape. The exclusive semileptonic decay B πlν determines the CKM matrix element V ub through the following formula, e e- o f e dγ dq = G F 4π 3 k π V ub f + (q ), (1) where the form factor f + is defined as 9 9

B πlν, quenched approximation f 0,+ (q ).0 1.0 UKQCD APE Fermilab JLQCD NRQCD LCSR LCSR Onogi, CKM 003. 0.0 0 5 10 15 0 5 30 q (GeV ) 10 10

B πlν, unquenched.5 N f =+1 (HPQCD) N f =+1 (FNAL/MILC) 1.5 f + f 0 Onogi, Lattice 006. Results agree well with quenched results. Probably not significant; not true for all quantities. 1 0.5 B!>πlν 0 0 5 10 15 0 5 q [GeV ] 11 11

B πlν, finite range of q Proposals to address:.5 N f =+1 (HPQCD) N f =+1 (FNAL/MILC) f + f 0 *) Moving NRQCD (Davies, Lepage, et al.) 1.5 1 0.5 B!>πlν *) Calculate in charm region, extrapolate to bottom (Abada et al.) F + 0 5 0 5 10 15 0 5 q [GeV ] 10 BABAR B πlν 0 0 10 0 q Lattice data extend over only a fraction of the q range on the physical B πlν decay. With standard methods, discretization errors go like O(ap), signal goes like exp(-eπt). *) Gibbons: global simultaneous fit of all experimental and lattice data. *) Unitarity and analyticity (Lellouch, Fukunaga-Onogi, Arnesen et al., Becher-Hill,...) 1 1

B πlν, unitarity fits 3 10 BABAR F + 5 P! F + 1 0 0 0 10 0 q P (t) φ(t, t 0 ) f(t) = -1-0. 0 0. -z a k (t 0 )z(t, t 0 ) k k=0 Vanishes at subthreshold (e.g. B*) poles Arbitrary analytic function -- choice only affects particular values of coefficients (a s) Pronounced q dependence in form factor is due to calculable effects. When those are factored out, two parameters suffice to describe the current experimental data. (Just like B Dlν, K πlν?!!) 13 13

B πlν, unitarity fits B->" form factor data normalized by P(t) x!(t,t 0 ) vs. z(t) P(t)!(t,t 0 )f(t) 0.1 0.05 P(t)!(t,t 0 )f 0 (t) P(t)!(t,t 0 )f + (t) Coefficients in z expansion are compatible with experiment. 0-0. -0.1 0 0.1 0. z(t) q max a0: a1: a: 0.0578 +- 0.003 0.0056 +- 0.068 0.1534 +- 0.41 14 14

B πlν, unitarity fits f 0 (q ) and f + (q ) 3.5 3.5 1.5 1 0.5 0 B->" semileptonic form factors vs. q 3 param. fit constrained such that f + (0)=f 0 (0) --! /d.o.f. = 0.35 f + (q ) from constrained fit unconstrained 3 parameter fit--! /d.o.f. = 0.35 f + (q ) from unconstrained fit 0 5 10 15 0 q (GeV ) Combined fits of f+ and f0 may give surprisingly good prediction for form factors well beyond the range of lattice data. - Raw lattice data, - Not extrapolated in m or a, - Momentum dependent discretization errors not yet included. How can the results of such fits best be compared with experiment? 15 15

Not covered, but interesting B ρlν, B ωlν, etc. Honest methods for treating unstable particles on the lattice exist (Lüscher,...) but they are much more demanding. B Klν, non-standard Model effects Lattice calculation are no more difficult as long as effective operators are local. 16 16

Summary and to-do list For lattice theorists: how well do lattice methods agree? staggered vs. clover vs. overlap, etc. Will Moving NRQCD allow calculation of the form factors for B πlν in the whole decay region? For theorists and experimentalists: how should lattice data be reported; how should lattice and experiment be compared? Raw lattice data in large global fit (Gibbons) Normalization and slope in the z expansion Form factor and slope at several fiducial points (Becher and Hill) All of the above 17 17